Solar Sails

Solar sails are the only method of spacecraft propulsion in which no fuel is needed. Until recently spacecraft powered by solar sails were the stuff of science fiction. However,  following the success of the Japanese spacecraft IKAROS which flew close to Venus in 2010, and in 2019 the crowd-funded Light Sail 2 spacecraft it is likely that spacecraft powered by solar sails will play an increasing role in future space exploration.

The Sunjammer a 1964 short story by Arthur C Clarke. It features the racing of space-yachts powered by solar sails.

IKAROS on its way to Venus in 2010 – image credit Wikimedia Commons

 

How do solar sails work?

Solar sails work by using radiation pressure from sunlight to produce thrust.  Effectively a solar sail is a mirror, reflecting sunlight that hits it.

Light consists of a stream of photons, whenever a photon hits the surface of a mirror it is reflected back at the same angle it strikes the mirror, shown as the Greek letter theta (θ) above. Some of the photon’s momentum is transferred to the mirror -pushing it away in a direction at right angles to its surface. The the net effect of the vast numbers of photons in sunlight is to generate a force or thrust accelerating  the mirror away from the Sun in the direction shown by the red arrow.

Because the direction of the thrust is at right angles to the mirror’s surface it is possible to change it by orientating the mirror at different angles to the Sun, making it possible to steer a solar powered spacecraft.

The diagrams show how the direction of the thrust changes with the solar sail’s orientation to the Sun.

The magnitude of the thrust also varies with the sail’s orientation. It is at its greatest when the sail is perpendicular to the Sun’s direction and falls off as the angle to the Sun decreases. When the sail is edge on, i.e. at an angle of 0^o to the Sun, the thrust is zero.

The table above shows thrust on a scale where the value when the mirror is perpendicular to the Sun is 100%. It assumes that all sunlight is reflected from the sail.

Light absorbed by the mirror

In reality, no mirror is 100% efficient, reflecting all light that hits it. Some light is always absorbed. The direction of the thrust caused by photons which are absorbed rather than reflected is always where the light is coming from regardless of the orientation of the mirror. This is illustrated below. In addition, the maximum thrust caused by absorbed light is only 50% of that caused by reflected light.

Variation with distance from the Sun

The intensity of sunlight, and therefore the thrust on a solar sail falls as the inverse square of the distance from the Sun.

Using solar sails as a method of propulsion

Force is measured in newtons (for those of you in the UK and the US who prefer imperial units 1 newton equals 0.225 pounds force).  At the Earth’s distance from the Sun, the intensity of sunlight a number known as the solar constant is 1.361 kilowatts per square metre.  Assuming that:

• all sunlight is reflected, (i.e none is absorbed by the sail) and
• the sail is perpendicular to the Sun,

this value of the solar constant provides a thrust of 0.000 009 08 newtons per square metre of sail. So, a sail having an area of 100 square metres would have a maximum thrust of 0.000 908 newtons (0.0002 pounds in imperial units).

This is a very weak thrust compared to traditional rockets. For example, the first stage of the SpaceX Falcon 9, which lifts the spacecraft off the launchpad, generates a massive 7.6 million newtons (1.7 million pounds) of thrust and burns for only 162 seconds before all its fuel is exhausted. The second stage, which produces 930 000 newtons of thrust, then kicks in and burns for a further 400 seconds to take the payload into orbit. Even the small thrusters on the SpaceX Dragon 2 space capsule, which fire for a short time to adjust its position each develop 400 newtons of thrust.

From Isaac Newton’s second law of motion.

Force (i.e. thrust on the spacecraft) = mass x acceleration (i.e. rate of change of velocity)

 So, the acceleration provided by a solar sail is its thrust divided by the mass of the spacecraft. To maximise acceleration the thrust must be as large as possible and the mass of the spacecraft as small as possible.

 A simple illustration

If we have a solar sail measuring 30 by 30 metres, giving a total area of 900 square metres and we assume that it is 90% efficient then its thrust is

900 x 0.000 009 08  x 0.9 = 0.0074 newtons

If we make the sail out of a material 0.01 mm thick having a density of 1 grammes per cubic centimetre (1000 kg per cubic metre) then the total mass of the sail (area x thickness x density) is

900  x 0.00001 x 1000 = 9 kg

However, in addition to the mass of the sail the spacecraft has its  own mass. If we assume that the total mass of the spacecraft and sail is 40 kg then the acceleration (thrust divided by mass) provided by the sail is a meagre

0.0074 newtons/ 40kg  = 0.000185 metres/second ²

 

However, the great beauty of solar sails is that they require no fuel. They provide a small continual acceleration in contrast to rockets which provide a powerful thrust for a short time (which is usually measured in minutes)

So, assuming there were no other forces acting on the spacecraft,  over a period of one day (86 400 seconds) the spacecraft could change its velocity by:

 86 400 x 0.000185  ≈  16 metres per seconds  ( which is roughly 58 km/h).

Although this is still a small change in velocity, over six months the spacecraft could increase its velocity by 10 600 km/h – without any fuel being used.

This illustrates the key feature of solar sails- they cannot produce the large thrusts needed to lift rockets off the ground, but the small acceleration mounts up over a period of time to produce large changes in velocity. This could make them suitable for powering unmanned spacecraft travelling  between planets.

Some more concrete examples

The illustration above is an oversimplification, because it assumes that the only force acting on the spacecraft is the solar sail thrust.

In reality, things are a little more complicated.

• The spacecraft will be in orbit around the Sun, and the force of the Sun’s gravity will pull it toward the Sun. The Sun’s gravity and the spacecraft’s initial orbital parameters (distance and starting velocity) must be taken into account when calculating its trajectory.
• If the spacecraft passes another massive object, e.g. a planet, then the effect of the gravitational force due to the object will need to be taken into account.
• The thrust provided by the solar sail and the gravitational force due to the Sun are continually changing as the  distance to the Sun varies.
• If the orientation of the solar sail to the Sun changes, both the magnitude and direction of the thrust will change.

 

Here are some example trajectories

Example 1

In this case, the spacecraft has a mass of 10 kg and starts off in a circular orbit at a distance of one astronomical unit (AU), which is 150 million km, from the Sun. It has a very large solar sail, inclined at an angle of 45 degrees to the direction of the Sun, which produces a thrust of 0.003 newtons.

Note that at the initial distance of 150 million km from the Sun the force of the Sun’s gravity on a 10 kg mass is 0.0593 newtons, roughly 20 times the thrust provided by the solar sail. Interestingly, because the force of the Sun’s gravity and the intensity of sunlight both fall as the inverse square of the spacecraft’s distance from the Sun, provided the sail maintains an orientation  of 45 degrees to the Sun’s direction, then the Sun’s gravity will always be twenty times larger than the thrust on the sail regardless of the spacecraft’s distance from the Sun.

Assuming the orientation remains at 45 degrees to the direction of the Sun, the spacecraft will gradually spiral away from the Sun. Over an 18-month period it will follow the trajectory shown below.

The numbers show the distance from the Sun in astronomical units. After 18 months the spacecraft will be approximately 1.6 AU from the Sun.

 

Example 2

This is the same as the previous example with the exception that the sail is orientation in such a way as to decelerate the spacecraft in its orbit, i.e the thrust is directed to cause the spacecraft to lose energy.

In this case assuming the orientation remains at 45 degrees to the direction of the Sun, the thrust provided by the sail will cause the spacecraft to steadily lose energy and it will spiral inwards towards the Sun. Over a 12-month period it will follow the trajectory shown below.

After 12 months the spacecraft will be only 0.56 AU from the Sun

 

Example 3

In this case, once again the spacecraft has a mass of 10 kg and starts off in a circular orbit at a distance of 1 AU from the Sun. This time the sail is orientated at an angle of 90 degrees to  the direction of the Sun and  produces a thrust of 0.006 newtons.

In this case the spacecraft’s orbit changes shape from circular to elliptical. Its closest distance from the Sun is 1 AU and its furthest is 1.25 AU. It takes 460 days to complete its new orbit.

Example 4

This is a modified version of example 1; as in all the previous examples, the spacecraft has a mass of 10 kg and starts off in a circular orbit at a distance of 1 AU from the Sun. However, this time, it has an extremely large solar sail, orientated at an angle of 45 degrees to the direction of the Sun. This produces a thrust of 0.03 newtons, – roughly half the strength of the Sun’s gravity.

In this case the thrust is strong enough for the spacecraft to rapidly escape the Sun’s gravity. Rather than gradually spiralling outwards, the spacecraft escapes from the solar system on a hyperbolic trajectory into interstellar space. After nine months the spacecraft will be 4.4 AU from the Sun.